Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution

Transcription

1 IEEE TRANS ON IMAGE PROCESSING Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution SG Roux () M Clausel () B Vedel (3) S Jaffard () P Abry () IEEE Fellow () Physics Dept ENS Lyon CNRS UMR567 Lyon France stephaneroux@ens-lyonfr patriceabry@ens-lyonfr () Laboratoire Jean Kuntzmann UMR 5 University of Grenoble France (3) LMBA University of Bretagne Sud European University of Bretagne Vannes France () University of Paris Est LAMA CNRS UMR 85 Créteil France Abstract Textures in images can often be well modeled using self-similar processes while they may at the same time display anisotropy The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes It will first be shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard Ddiscrete wavelet transform by the hyperbolic wavelet transform which permits the use of different dilation factors along the horizontal and vertical axis Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized this discrepancy defines a rotation angle Second we show that this rotation angle can be jointly estimated Third a non parametric bootstrap based procedure is described that provides confidence interval in addition to the estimates themselves and enables to construct an isotropy test procedure that can be applied to a single texture image Fourth the robustness and versatility of the proposed analysis is illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields As an illustration we show that a true anisotropy built-in selfsimilarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed Index Terms Self-Similarity Anisotropy Gaussian Fields Hyperbolic Wavelet Transform Scale Invariance Rotation Invariance Anisotropy Test Bootstrap I INTRODUCTION Texture classification In numerous modern applications (satellite imagery [] geography [] biomedical imagery [3] [7] geophysics [8] art investigation [9] ) the data available for analysis consist of images of homogeneous textures that need to be characterized Texture classification thus consists of classical problem in image processing that received considerable efforts in recent years (cf eg [3] [] [6] and references therein) Scale invariance and Self-similarity Amongst the many different ways texture characterizations have been investigated techniques based on scale invariance or fractal concepts are considered as promising notably for the application fields listed above (cf eg [7] for a review) Scale invariance can be defined as the fact that there exists no specific space-scale in data that play a preferred role in their space dynamic or equivalently that all space-scales are equally important Scale invariance in data implies that they are analyzed with (statistical) models that do not rely on the identification of specific scales (such as Markov models) but instead with models that aim at characterizing a relation amongst scales Because Self-Similarity is a theoretically well-grounded and relatively simple instance of scale invariance behaviors it has often been proposed that Gaussian self-similar fields are relevant models enabling efficient characterization and classification of the textures (cf eg [5] [6]) Anisotropy However textures are also often characterized by anisotropy which may either be deeply tied to selfsimilarity itself [8] [9] or exist as an independent property that is superimposed to an isotropic self-similarity In both cases it is a crucial stake in analysis to disentangle self-similarity from anisotropy to discriminate whether self-similarity and anisotropy are independent properties or if they are stemming from the same constructive mechanism as well as to be able to estimate accurately the self-similarity parameter H despite anisotropy It has already been pointed out that fractal analysis and estimation is very sensitive to anisotropy (cf eg [8]) In the literature anisotropy is often analyzed from D slices extracted from images along different directions [6] or by making use of local directional differential estimators [] [] Goals and contributions In this context elaborating on a preliminary attempt [] the present contribution aims at proposing an efficient and elegant solution to the joint analysis and estimation of self-similarity and anisotropy in D fields Though the devised procedure aims at and is designed for being applied to real-world textures its performance are assessed by means of Monte Carlo simulation performed on synthetic isotropic and non isotropic Gaussian textures While the (discrete) wavelet transform (DWT) is nowadays a classical tool for image processing the key originality of the present work is to show that the classical discrete wavelet transform fails at providing a relevant analysis of self-similarity in presence of anisotropy and it is instead here proposed to replace it with the Hyperbolic Wavelet Transform (HWT) (defined eg in [3]) Indeed the use of different dilation factors on the axes

2 IEEE TRANS ON IMAGE PROCESSING x and y potentially permits to see the anisotropy as opposed to the classical D Discrete Wavelet Transform relying on a single and isotropic dilation operator The Hyperbolic Wavelet Transform is defined in Section II-B Note that the HWT had appear earlier in the literature under different names such as Tensor-product wavelet [] anisotropic wavelet transform [5] or rectangular wavelet transform [6] without specific exploration though of its benefits to study anisotropy in textures Also redundant (or overcomplete) wavelet representations (such as M-band dual tree and Hilbert pair complex wavelets cf eg [7] for a enlightening review) may be used to analyze images and textures However while they suffer from a larger computational cost they have been observed (in preliminary attempts performed by the authors) to yield little if not no practical benefit for the study of scale invariance and the estimation of the corresponding parameters Overcomplete wavelet representations are thus excluded from the present study As representative of D model mixing self-similarity and anisotropy self-similar Gaussian D random fields with builtin anisotropy such as those proposed in eg [8] [9] are used here These processes are defined and illustrated in Section II-A and their hyperbolic wavelet analysis is detailed in Section II-C Estimation procedures for the parameters characterizing self-similarity and anisotropy are defined and their performance assessed in Section III-A The definition of anisotropy involves a rigid definition of reference (orthogonal) axes that have no reason a priori to match those of the sensor used to acquire the image and thus to coincide with the horizontal x and vertical y axis with which the image is presented for analysis Therefore the model introduced in Section II-A includes a rotation parameter that accounts for this unknown An estimation procedure for this rotation is devised and analyzed in Section III-B Therefore the parameters characterizing rotation anisotropy and self-similarity are estimated jointly For application purposes it is crucial to be able to decide whether textures should be modeled by isotropic or anisotropic models Therefore a procedure for testing the null hypothesis that the texture is isotropic is constructed and studied in Section IV It is based on a non parametric bootstrap procedure performed on the hyperbolic wavelet coefficients (in the spirit of the construction devised in [8]) and can thus be applied to each single analyzed texture independently Incidentally the bootstrap procedures also provides us with confidence intervals for the estimates a very important feature for practical purposes To finish with the analysis procedures proposed here are applied in Section V to a variety of isotropic and anisotropic fields that differ from the precise model used as a reference model (cf Section II-A) hence illustrating the robustness and generality of the tools proposed here Notably it is shown that the proposed analysis enables to clearly distinguish between a truly anisotropic self-similar field from a isotropic self-similar field (with same self-similar parameter) to which directional hence anisotropic oscillations have been additively superimposed Joint anisotropy and self-similarity is hence clearly disentangled from isotropic self-similarity with Fig Sample fields of X θ α H Top line : (θ α H ) = ( α ) with from left to right α = (isotropic); α = 7 and α = 3 Bottom line : (θ α H ) = (θ 7 ) and from left to right θ = /6 θ = / and θ = /3 unrelated superimposed anisotropic trend II HYPERBOLIC WAVELET ANALYSIS OF ANISOTROPIC SELF-SIMILAR RANDOM FIELDS A Anisotropic self-similar random fields ) Definition: Because of its generic and representative nature it has been chosen to work with the class of anisotropic Gaussian self-similar fields introduced in [8] [9] referred to as Operator Scaling Gaussian Random Field (OSGRF) which can be defined using the following harmonizable representation: X feh (x) = (e i x ξ )f(ξ) (H+) dŵ (ξ) () R where x = (x x ) ξ = (ξ ξ ) E is a matrix satisfying Tr(E ) = f a E homogeneous continuous positive function (hence satisfying the homogeneity relationship f(a E ξ) = af(ξ) on R ) such that ( ξ )f(ξ) (H+) dξ < + and where dŵ (ξ) stands for a D Wiener measure When f is not a radial function the Gaussian field is not isotropic In this study it is chosen to use the following - parameter (related to anisotropy and rotation) explicit form: f θα (ξ) = ( ζ /α + ζ /( α) ) with ζ = (ζ ζ ) = R θ ξ and rotation matrix R θ defined as: ( ) cos(θ ) sin(θ R θ = ) sin(θ ) cos(θ ) ( ) α In this model E = < α α < A rotation parameter θ has been added to the definition : It accounts for the fact that the rigid axes according to which anisotropy is defined need not match a priori the sensor axes according to which the image is digitalized (for realworld data) or numerically produced (for synthetic textures) In the sequel OSGRF X θα H thus refers to the following model relying on 3 parameters θ α H characterizing respectively rotation anisotropy and self-similarity X θα H (x) = (e i x ξ )f R αθ (ξ) (H+) dŵ (ξ)()

3 HYPERBOLIC WAVELET TRANSFORM FOR SELFSIMILAR ANISOTROPIC IMAGE ANALYSIS 3 ) Properties: With this construction OSGRF X θα H has stationary increments It possesses a built-in anisotropy characterized by the parameter α ( ) When α = the field is isotropic and the case < α < correspond to the case < α < with the axes (x x ) permuted OSGRF X θα H satisfies (where = L denotes equality for all finite dimensional distributions): (LL) (HL) (LH) (HH) (LL) (HLH) (LLHH) (LHH) (HH) {X αh θ (a E x)} L = {a H X αh θ (x)} (3) with E = R θ ER θ It is thus exactly self-similar with parameter < H < min (α α ) < Fig displays realizations of X θα H (x) obtained from MATLAB routines written by ourselves and available upon request On top row (H θ ) = ( ) are kept fixed while α is varied from α = 7 and 3 (from left to right) The practical goal is to estimate correctly H despite these different unknown anisotropy strengths α On bottom row a strongly anisotropic field is shown (α = 3 H = ) with 3 different rotation angles θ (from left to right θ = /6 / and /3) The targeted goal is here to estimate correctly (α H ) despite such unknown rotation This three-parameter OSGRF stochastic process provides us with a rich and versatile model for selfsimilar (an)isotropic textures 3) Numerical simulation: Realizations (or sample fields) of the synthetic processes defined in Eq () are produced numerically following the classical procedure recalled in eg [9] relying on drawing at random realizations of whitenoise dŵ (ξ) followed by standard numerical integration procedures B Hyperbolic Wavelet Transform The D Hyperbolic Wavelet Transform (HWT) differs from the D Discrete Wavelet Transform (DWT) insofar as its definition relies on the use of two different dilation factors along the horizontal and vertical axes as opposed to the D- DWT that makes use of a single and same dilation factor along both axes This difference turns out to be crucial for the analysis of anisotropy The collection of functions constituting the orthogonal basis underlying the HWT is defined as tensor products of univariate wavelets (cf eg [9]) Let ϕ and ψ denote the scaling function and the wavelet of a given one-dimensional multiresolution analysis The HWT basis of L (R ) is defined as (cf [3]): ψ jj k k (x x ) = ψ( j x k )ψ( j x k ) ψ jk k (x x ) = ϕ(x k )ψ( j x k ) ψ jk k (x x ) = ψ( j x k )ϕ(x k ) ψ kk (x x ) = ϕ(x k )ϕ(x k ) for all j = (j j ) N and k = (k k ) Z The HWT shares a deep relation with Triebel bases used in mathematic literature to characterize anisotropic functional spaces (cf [3]) (LLLL) (LHLH) (HH) log (fj /3) log (f j /3) Fig Hyperbolic Wavelet Transform Top line : one step of HWT consists of one step of D-DWT (left) (with D-DWT performed on each line of HL and each column of LH subbands (right)) Bottom line : second step of HWT (left) and locations of the HWT vs DWT (black dots) coefficients in the frequency domain (right) Black dots correspond to D-DWT while for HWT white dots indicate subband HH black triangles indicate LH and HL squares correspond to the approximation coefficients) The circle symbols correspond to the coefficients ψ j j j j and square symbols to the coefficients ψ j ψ j ψ The hyperbolic wavelet coefficients of the process X are defined j = (j j ) N as : d X (j k) = j+j R ψ jk (x x )X(x x )dx dx () Note that a L -normalization is used (instead of the classical L -norm) as it better suits self-similarity analysis (cf eg [3]) With these notations fine resolution scales correspond to the limit j j + and index j in the decomposition corresponds to the actual resolution J j where J = log (N) for an image of size (N N) Such coefficients d X (j k) can be computed efficiently using a recursive pyramidal filter bank based algorithm comparable to that underlying the D-DWT In Fig the first two iterations are illustrated in the Fourier domain One iteration of HWT practically consists of the combination of one iteration of the D-DWT algorithm with D-DWT performed on each line of the vertical details (HL) and D DWT performed on each column of the horizontal details (LH) Because the central frequencies of the dilated scaling function of ϕ( j x) and mother-wavelet ψ( j x) can be approximated as f j = j and f j = 3 j respectively the HWT coefficients d X (j k) can be located in a (log-) frequency-frequency plane as shown in Fig bottom right) and thus compared to the location of the D-DWT coefficients In what follows a D-Daubechies-3 multiresolution is used [3]

4 IEEE TRANS ON IMAGE PROCESSING C Analysis of anisotropic self-similar random fields From Eq () the HWT coefficients can be rewritten j = (j j ) N and k = (k k ) Z as stochastic integrals: ( ) l= ei j l k l ξ l ψ( j lξl ) d X (j k) = ( ξ /α + ξ /( α)) H dŵ (ξ + ξ ) R (5) Following the methodology in [33] [3] it can be proven that the HWT coefficients are weakly correlated ie (j j k k k k ) E(d X ((j j ) (k k ))d X ((j j ) (k k ))) E( d X((j j ) ) + k k + k k (6) Using the substitution ζ = j ξ ζ = j ξ in the rewriting of definition of the wavelet coefficients (cf Eq (5)) we have been able to show that the HWT coefficients typically behave as [33] [3] : d X (j k) j +j R ( ) l= eik lζ l ψ(ζl ) dŵ (ζ ζ ) ( j α ζ α + j ) H+ α α ζ When j /α > j /( α ) we derive that for all ζ ζ : α ( j α j α ζ α j α ζ α + j α ζ and further j H α ( ζ α + ζ α ) H+ j α ζ α + j α ζ α ζ α + ζ ) α jh α ( ζ H + α ) which enabled us to obtain the following inequality: C j +j j (H +) α E( d X (j k) ) / C j +j j (H +) α ( l= with C = ψ(ζ ) / l ) dξ R ( ζ α α + ζ ) H+ ( l= and C = ψ(ζ ) / l ) dξ R ( ζ H + α ) Combined to similar arguments for the case j /α j /( α ) these calculations enable us to show that the order of magnitude of the expectation of the squared HWT coefficients reads (j j ): E( d X (j k) ) / j +j (H+) max( j α j α ) (7) Because the processes of interest here are Gaussian this can straightforwardly be extended to any q cf [33] [35]: E( d X (j k) q ) q(j+j) q(h+) max( j α j α ) (8) These key results constitute the founding ingredient for the estimation procedures defined below j j S(q = j j ) (d) 3 5 j log (S(q = αj ( α)j)) τ(q α)/q Ĥ 6 6 ˆα (e) 6 Ĥ (f) 6 ˆα 3 j 5 5 α 5 Fig 3 Illustrations of the estimation procedures For anisotropic α = 7 (top row) and isotropic α = (bottom row) OSGRF X θ α H with θ = H = 7 Left column: Structure functions S(q j j ) The solid line indicates the direction ˆα while the dashed lines corresponds to α = and hence the sole direction actually reachable with the coefficients of D-DWT Middle column: Estimation of H(α) based on τ(q = α = ˆα) obtained from a linear regression of log S(q αj ( α)j) versus j ( ) Solid lines correspond to the theoretical τ(q = α ) Stars (in ) correspond to the (biased) estimation of H from τ(q = α = ) ie by using only the D- DWT coefficients Right column: Plots of Ĥ(α) = τ( α)/ versus α The black dot shows the location of the maximum of H (α) = τ( α)/ thus yielding the estimated ˆα and Ĥ In (f) as expected for an isotropic image ˆα = The mixed line corresponds to the theoretical values of τ( α)/ (cf Eq ) III PARAMETER ESTIMATION The goal is now to define estimation procedures for the three-parameters entering the definition of OSGRF X θα H and to study their statistical performance It is assumed first that θ is known and equal to θ and estimation is devoted to parameters α and H In the second part θ is unknown and needs to be estimated as well This is performed by applying the estimation of α and H to a collection of rotated images It will be shown that the correct estimation angle is estimated when the estimation of the anisotropy coefficients reaches its minimum A Self-similarity and anisotropy parameters In this section θ is assumed to be known and taken equal to for simplicity ) Estimation procedure: By analogy to what has classically been done for the analysis of self-similarity or scale invariance in general (cf eg [36]) the space averages at joint scales (j j ) (also referred to as structure functions) are used as estimators for the ensemble averages appearing in Eq (8) above: S(q j j ) = n jj (k k ) Z d X (j j k k ) q (9) where n jj stands for the number of available coefficients jointly at scales ( j j ) Let us further define τ j (q α) as a function of the statistical order q > and of the anisotropy parameter α: τ(q α) = lim inf j α log (S(q αj ( α)j) () j

5 HYPERBOLIC WAVELET TRANSFORM FOR SELFSIMILAR ANISOTROPIC IMAGE ANALYSIS 5 In essence Eq () amounts to assuming a power-law behavior of the structure functions with respect to scales in the limit of fine scales j + along direction α: τ(qα) j S(q αj ( α)j) S (q) α Eq (8) above indicates that on average and with the specific choice (j j ) = (αj ( α)j): ( ) S(q αj ( α)j) j q α (H +) max( α α α α ) τ(qα) α = Comparing ( these two last relations ) suggests that q α (H + ) max( α α α α ) so that for a given fixed q the anisotropy parameter α can be estimated as: ˆα q = argmin α τ(q α) () and that the self-similarity parameter H can be estimated as: Ĥ q = τ(q ˆα q )/q () ) Illustrations: The estimation procedure proposed here is sketched in Fig 3 for q = for an anisotropic (3a-c) and an isotropic (3d-e) OSGRF X θα H It can be decomposed into three steps (for a given q > ) Step : The HWT coefficients d X (j k) and corresponding structure functions S(q j j ) are computed Examples are shown in Fig 3 (left column) Step : The surface log S(q j j ) seen as a function of the variables j j is interpolated (by nearest neighbor) along the line αj + = ( α)j + Then a non weighted leastsquare regression of log S(q αj ( α)j) versus log j = j is performed across all available scales hence yielding an estimate of τ(q α) for each α and each q as sketched in Fig 3-b and 3-e) Step 3: The estimated τ(q α) are plotted for a given q as a function of α and its maximum yields the estimate ˆα of α (cf Fig 3-c and 3-f) The estimation of the self similarity parameter H is further given by Ĥq = τ(q ˆα)/q This procedure calls for the following comments First Step is performed for all accessible αs that is for all values of α that connect at least two pairs of dyadic scales ie (a = j a = j ) with integers (j j ) = [ J] Therefore the actual resolution of the values of α that can actually be used depends on the size N N of the analyzed image and hence so does the resolution of the estimate of the anisotropy parameter This discretized resolution can be observed in Fig 7 Second the structure functions S(q j j) (hence for α = ) are computed from HWT coefficients that actually corresponds to those of the D-DWT (cf Fig 3-b dashed line) For isotropic fields it is found that ˆα = and thus that the coefficients of the HWT that need to be actually used for the estimate of H are those of the D-DWT Conversely for anisotropic fields basing the estimate of H on S(q j j) results into significant biases as illustrated in Fig 3-b dashed line and -e where the estimate of H for α significantly differs from that obtained for α = ˆα This illustrates the major benefits of replacing the D-DWT with the D-HWT EĤ H log (σ Ĥ ) log (N) Eˆα α log (σˆα) 6 8 (d) log (N) Fig Estimation performance As functions of the sample size N (image size N N) biases (top row) and standard deviation for Ĥq= (left column) and ˆα q= (right column) obtained as average of estimation performed on 5 realizations of OSGRF X α H with parameters α = H = 7 5 and 3 ( ) α = 8 H = ( ) and α = 6 H = 5 3 ( ) Bottom row dashed lines illustrates the expected /sqrtn N decrease of the standard deviations 3) Estimation Performance: To complement the theoretical study reported above and to further assess the performance of the proposed estimation procedures Monte-Carlo simulations are now performed further expanding on numerical investigations presented in [] Biases and the standard deviations of ˆα q and Ĥq are obtained from averages of estimates computed over 5 independent realizations of OSGRF X αh numerically produced by MATLAB routines designed by ourselves and available upon request Fig reports biases and standard deviations as a function of (the log of) the sample size N (image size is N N) for various parameters (α H ) Fig essentially shows that the estimation performance both for ˆα and Ĥ does not depend on the actual H a result that is highly reminiscent of the D case (cf eg [37]) However dependences on the anisotropy parameter α do exist and are clearly visible on the standard deviation which unexpectedly decrease with significant departures of isotropy Estimates are found to be asymptotically unbiased as expected from theoretical analysis and that standard deviations roughly decrease as / (N N) = /N in agreement with the weak correlation property of the HWT coefficients Using other values of q > (ranging from to 5) yields similar conclusions To conclude this section let us put the emphasis on the fact that image sizes are varied from small ( 8 8 ) to (very) large ( 3 3 ) This illustrates that both the synthesis and analysis procedures corresponding to the definition of OSGRF and its analysis can be implemented efficiently and benefits from a remarkably low computational cost Estimation performance were reported here only for q = as it was found empirically that the use of other values of q

6 6 IEEE TRANS ON IMAGE PROCESSING ˆα Ĥ 3 (α = 7 H = 6) 8 (α = 3 H = ) 3 (α = 7 H = ) (d) θ (e) θ (f) Fig 5 Joint three parameter estimation procedure Estimations of ˆα (top row) and Ĥ (bottom row) for three differents fields with θ = /3 and (α H ) = (7 6) ( and (d)); (α H ) = (3 ) ( and (e)); (α H ) = (7 ) ( and (f)) The dashed line illustrates the expected theoretical behavior of ˆα as a function of θ and the circle with confidence intervals to Monte-Carlo averages The estimation of ˆθ corresponds to the location of the minimum of ˆα(θ) and satisfactorily corresponds to θ = /3 Final estimates for α and H are obtained as ˆα = ˆα(ˆθ) and Ĥ = Ĥ(ˆθ) and thus show satisfactory agreement with the theoretical values marked by Error bars correspond to σĥ (resp σ ˆα ) did not improve performance as can be expected for Gaussian processes B Rotation parameter ) Estimation procedure: Let us now consider the case where in addition to H and α the rotation angle θ is unknown To estimate jointly the three unknown parameters it is here proposed to apply the above procedure to estimate H and α to a collection of rotated version of the original image with rotation angles θ The estimation of the anisotropy direction relies on the following observations illustrated in Fig 5 (top row): i) The estimate ˆα(θ) is a -periodic function ; ii) it also has the symmetry ˆα(θ + θ) = ˆα(θ θ) ; iii) when θ = θ ˆα α ; iv) when θ = θ + / ˆα = α ; v) and when θ = θ + / ˆα = Thus the following joint estimation procedure for (θ α H ) can be proposed: θ ˆθ = argmin θ ˆα(θ) (3) ˆα = ˆα(ˆθ) () Ĥ q = Ĥq(ˆα(ˆθ)) (5) Because the minimum of ˆα(θ) is (arbitrarily) picked this procedure necessarily implies ˆα there is thus a remaining indetermination whether the correct choice is ˆα or ˆα and therefore of / in θ As previously mentioned this only amounts to exchanging the roles of the axis x and y For isotropic fields ˆα(θ) fluctuates around α = and no clear minimum (or maximum) is visible Furthermore ˆθ max ˆθ min differs from / When ˆθ max ˆθ min < / the field is thus declared isotropic and we set ˆθ = To practically perform the rotation of θ on the image of analysis a nearest neighbor interpolation is applied The procedure is totally automated and no human supervision is needed 3 EσBS(Ĥ)/σ MC(Ĥ) H EσBS( ˆα)/σMC( ˆα) H Fig 6 Bootstrap versus Monte Carlo Estimates of standard deviations Estimations of Eσ BS(Ĥ)/σMC(Ĥ) as a function of H for q = obtained from R = bootstraps applied to independent copies of OSGRF X α H (image size ( )) with parameters α = ( ) α = 8 ( ) and α = 6 ( ) Estimations of Eσ BS(ˆα)/σ MC(ˆα) as a function of H for α = ( ) α = 8 ( ) and α = 6 ( ) ) Illustrations and performance: To assess the performance of the joint three-parameter estimate procedure Monte- Carlo numerical simulations are conducted and biases and standard deviations are computed from average over realizations of OSGRF X θα H for various choices of (θ α H ) and with q = Fig 5 shows top row that the estimation ˆα(θ) clearly follows a piecewise linear variation along θ (modeled by the dashed line) and displays clear extrema for θ θ and θ θ + / For θ θ ˆα(ˆθ) and Ĥ(ˆθ) (bottom line of Fig 5) provide satisfactory estimates of α and H For θ = ˆθ+/ the estimations are ˆα and Ĥ Table I displays the biases standard deviations and Mean square errors for several isotropic and anisotropic OSGRF X θα H fields It can be observed that Ĥ shows more bias when a rotation of the original image is performed This is likely due to the interpolation procedure that smoothes out data and thus that distorts self-similarity and thus scale invariance at the finest scales Better estimations for H can be achieved by discarding a few of the finest scales from the linear regression when image size permits IV BOOTSTRAP-BASED ANISOTROPY TEST AND CONFIDENCE INTERVALS In applications it is often of crucial importance to be able to test the isotropy assumption (ie whether α = or not) for each single image independently This theoretically requires the knowledge of the distribution of ˆα Though it is found empirically Gaussian the variance of the distribution remains unknown and as suggested in Section III-A and Fig it depends not only on the sample size N but also on the unknown parameter α itself Asymptotic Gaussian expansions for the calculations of the theoretical variance of ˆα in the spirit of those proposed for fractional Brownian motion in eg [38] have been observed to perform poorly (not reported here) Instead it is proposed to apply non parametric bootstrap procedure in the HWT coefficient domain in the spirit of the procedures developed and assessed in [8] [39] [] This procedure is detailed in the next section while the corresponding bootstrapped based isotropy test is defined and assessed in Section IV-C

7 HYPERBOLIC WAVELET TRANSFORM FOR SELFSIMILAR ANISOTROPIC IMAGE ANALYSIS 7 A Bootstrap resampling schemes In a nutshell nonparametric bootstrap makes use of available samples many times by a drawing with replacement procedure to yield an approximation of the unknown population distribution In turn this estimated population distribution is used to construct confidence intervals or test (cf eg [] and [3]) For the present work following [8] the resampling procedure is applied in the HWT coefficient domain Because HWT coefficients do not consist of independent random variables but possess a residual correlation a time-block bootstrap procedure is used: At each octave j block of size l of HWT coefficients are drawn randomly with replacement This yields a set of bootstrapped HWT coefficients d X (j k) from which bootstrap estimates ˆα and Ĥ of α and H respectively are obtained This procedure is repeated R times and the population distribution of ˆα and Ĥ are inferred from the boostrap estimates ˆα r and Ĥ r r = R notably variances can be estimated B Bootstrap-based estimates of variance It has been found empirically that l need not depend on octave j and can be kept small As documented in [8] l is set to twice the size of the support of the mother wavelet (eg for a Daubechies3 wavelet used here l = 6) as correlations amongst HWT coefficients is found to remain significant essentially over a space-scale controlled by the size of the wavelet support Fig 6 compares the standard deviations of Ĥ (left) and ˆα obtained from Monte Carlo simulations for anisotropic fields (of size ) against those obtained by the bootstrap procedure (with R = for each of the Monte Carlo simulations) Fig 6 shows that the ratios σ BS (Ĥ)/σMC(Ĥ) and σ BS (ˆα)/σ MC (ˆα) depend neither on α nor on H and remain close to with a slight overestimation (from to %) for the former and quasi perfect match for the latter Equivalent conclusions are drawn from different sample sizes N These results indicate that the bootstrap estimates of the variances provide valuable approximations of the true variances of Ĥ and ˆα Together with the Gaussian distribution empirical fact this yields very satisfactory confidence intervals for Ĥ and ˆα C Test procedure and performance ) Test procedure: To test isotropy in a given image the null and alternative hypothesis respectively read: H : α = and H A : α (6) Let us assume first that θ The test procedure can be decomposed as follows: - Estimate ˆα as proposed in Section III-A - Apply the resampling scheme described in Section IV-A above to the HWT coefficients of X θα H and construct the bootstrap distribution estimate of ˆα from the boostrap estimates ˆα r r = R - Set the test significance level δ for the test P (ˆα) P (ˆα) ˆα Power of test β α Fig 7 Anisotropy test a) Histogram of ˆα MC (light gray) and of ˆα BS (black) for OSGRF X α H (image size ( )) with α = 75 and α = Right plot shows the rejection level of the test (with a significance level of 9%) obtained for R = bootstraps averaged on realizations of X α H with H = 3 ( ) 5 ( ) and 7 ( ) - Because when θ there is no reason to decide a priori that the true α will depart from by being larger or smaller a bilateral symmetric test is constructed Assuming a normal distribution for ˆα the bootstrapbased standard deviation estimation σ is used to construct the equi-tailed and symmetric acceptance region [ t δ/ σ t δ/ σ ] where t δ/ denotes the δ/-th quantile of the zero-mean unit variance Gaussian distribution - Alternatively the p-value of the test can be measured as the minimum between P (ˆα < ˆα(ˆθ)) and P (ˆα > ˆα(ˆθ)) divided by ) Test performance: To assess the validity and performance of the proposed test it has been compared against Monte Carlo simulations based on independent copies of OSGRF X θ=α H with various parameter settings and for image size Fig 7a) and 7b) compare the histograms of the estimates of α stemming from Monte Carlo simulations against those obtained from bootstrap estimates ˆα from a single realization chosen arbitrarily for anisotropic and isotropic fields For both cases distributions are found to be in satisfactory agreement These figures also show that ˆα can only take discretized values because of the finite sample size of the image as discussed in Section III-A In Fig 7c) the significance level of the test has been arbitrarily set to δ = 9 and the rejection level of the bootstrap test (R = ) has been computed as average over independant Monte Carlo realizations of OSGRF X θ=α H for various parameter settings When α = OSGRF is isotropic and the rejection level β is as expected found to satisfactorily reproduce the prescribed significance level δ = : ˆβ = 3 5 and respectively for H = 7 5 and 3 When α OSGRF is anisotropic and the rejection level β measures the power of the test Interestingly it is found that the estimated power does not depend on H is symmetric for α above and below and mostly that it increases sharply when α departs from This is thus indicating a strong potential to detect anisotropy even for small departure of α from 3) Test procedure for θ : When θ is unknown and needs to be estimated the procedure to test isotropy must be slightly amended as follows: - Apply estimation procedure for θ α H as in Sec-

8 8 IEEE TRANS ON IMAGE PROCESSING (θ α H ) (/3 7 6) (/3 7 ) (/3 3 ) ( 6) ( ) ˆθ θ ˆα α Ĥ H (stdmse) (stdmse) (stdmse) - (3) () () -5 (3) () () - -9 () (5) (9) 7 - (3) (8) (3) (5) (8) (3) % rej 8 H 7 H OSGRF EFBF FBS TABLE II Isotropie test: Rejection rates OBTAINED FOR THREE DIFFERENT CLASSES OF PROCESSES (FROM R = BOOTSTRAPS ON EACH OF THE REALIZATIONS SIGNIFICANCE LEVEL OF δ = 9%) TABLE I BIASES STANDARD DEVIATIONS AND MEAN SQUARE ERRORS OBTAINED FROM INDEPENDENT COPIES OF OSGRF X θ α H (IMAGE SIZE ( )) THE RIGHT COLUMN REPORTS THE CORRESPONDING REJECTION RATE OF THE ANISOTROPY TEST DESCRIBED IN SECTION IV-C3 WITH R = BOOTSTRAP SURROGATES THE SIGNIFICANCE LEVEL IS SET TO δ = % tion III-B - For ˆθ store the estimate ˆα(ˆθ) and the rotated field Xˆθ - Apply the resampling scheme described in Section IV-A above to the HWT coefficients of Xˆθ and construct the bootstrap distribution estimate of ˆα from the boostrap estimates ˆα r r = R - Set the test significance level δ for the test - Because the estimated ˆα necessarily takes values in [ ] a monolateral test must be constructed and the acceptance region is thus defined as: [ t δ σ ] - Alternatively the p-value of the test can be computed as P (ˆα < ˆα(ˆθ)) Table I (right column) reports the rejection rates of the procedure applied to several anisotropic and isotropic OSGRF X θα H fields of size ( ) For isotropic cases the rejection rates matches closely the significance level as expected For anisotropic fields the power of the test is found very high as soon as α departs even slightly from V OTHER ISOTROPIC AND ANISOTOPIC RANDOM FIELDS So far the analysis (estimation and test) procedures proposed here were applied only to the OSGRF X θα H defined in Section II-A and chosen as a convenient reference model with three parameters accounting jointly for rotation (an)isotropy and self-similarity However one can naturally wonder whether the isotropy test described above would satisfactorily perform to detect anisotropy for other models ie whether ˆα = or not In this section a number of isotropic and non isotropic self-similar models commonly encountered in the image processing and statistics literature are used to test the level of generality of the approach proposed here A Random fields ) Another OSGRF: In [] another interesting instance of OSGRF has been explored It is defined from Eq () with: f(ξ) = ( ξ + ξ a ) β (7) where β = H + ( + /a)/ and a = H /H for < H < H < This process resembles OSGRF X θα H in Eq () with α = a/( + a) H = ah /( + a) and θ = It is thus anisotropic as soon as a ) Extended Fractional Brownian Fields: Another class of possibly anisotropic Gaussian field referred to as Extended Fractional Brownian Filed was first introduced in [8] Its definition X f (x) = (e i x ξ )f(ξ) / dŵ (ξ) relies on R an admissible function f of the form: f(ξ) = ξ h(arg(ξ)) (8) where arg(ξ) is the direction of the frequency ξ and h an even measurable periodic function taking values in ( ) Fractional Brownian field is a particular and isotropic case of EFBF where h is a constant function but EFBF is in general anisotropic when h is not constant function Strictly speaking EFBF is not exactly selfsimilar (except in cases where h is a constant function) However EFBF shows scale invariance properties that are empirically close to those of strictly selfsimilar fields Fig 8a) shows a sample field of anisotropic EFBF with h(arg(ξ)) = H (cos( arg(ξ)) + ɛ) /( + ɛ) (9) where ɛ = + H /(H H ) Function h is -periodic and takes values in [H H ] Fig 8c) shows one sample-field obtained with parameters H = and H = 8 3) Fractional Brownian Sheet : Fractional Brownian Sheet (FBS) introduced in [5] provides us with another class of (an)isotropic self-similar Gaussian field It can be defined through its harmonizable representation for any (H H ) in ( ) (see [6]) : B HH (x) = (e i<xξ> )(e i<xξ> ) R ξ H+ ξ H+ dŵξ ξ () where dw xx is a Brownian measure on R and dŵξ ξ its Fourier transform FBS is a Gaussian field with stationary rectangular increments satisfying the following scaling property (a a ) (R +) {B HH (a x a x )} L = {a H ah B H H (x x )} () B Testing anisotropy The estimation and test procedures described above were applied to these three classes of processes for various setting of [H H ] Estimated function Ĥ(α) averaged over realizations (size ) are reported in Fig 8 right column Isotropy rejection rates obtained from R =

9 HYPERBOLIC WAVELET TRANSFORM FOR SELFSIMILAR ANISOTROPIC IMAGE ANALYSIS 9 5 Ĥ(α) 5 (e) Ĥ(α) Ĥ(α) (d) 6 8 (f) ˆα 8 θ 3 Ĥ 8 6 (d) Fig 9 Sine wave of direction θ = /8 with an isotropic field (H = 5 α = ); anisotropic self-similar fields with (H = 5 α = 6 θ = /8) ˆα and Ĥ (d) versus the angle analysis θ The symbols ( ) represent the results obtained for the field in and the ( ) for the field in θ α Fig 8 Other (an)isotropic selfsimilar Gaussian fields Left column sample fields with (H H ) = ( 8) for OSGRF (top) EFBF (middle) FBS (bottom) Right column Ĥ(α) obtained for averages over realizations with H = 8 and H = 8 ( ) 6 ( ) ( ) and ( ) Ĥ(α) clearly shows a maximum for α when fields are anisotropic bootstrap surrogates for each of the realizations are reported in Table II For the 3 class of processes when H H it is observed that Ĥ(α) has a maximum for α that clearly departs from and simultaneously that the isotropy rejection rates is far larger than the chosen δ = % significance level of the test This is the case even for as small discrepancies between H and H as H H = These results clearly show that the proposed procedures clearly detect anisotropy For EFBF it is reported in [] that the test anisotropy proposed therein failed to detect anisotropy (ie test reject in % of cases) when H = 5 and H = 7 Trying as careful a comparison as possible using the same model and parameter setting it is found that the bootstrap test described in Section IV-C yields rejection of isotropy with the δ = % significance level in % of cases hence showing a much improved power (cf Table II) Conversely for H = H it is observed for EFBF and OSGRF that Ĥ(α) has a maximum for α = and simultaneously that the isotropy rejection rates reproduces the targeted significance level hence confirming that these processes are isotropic For FBS Ĥ(α) remains flat for all αs while the rejection rates are higher than the targeted significance level this is thus questioning isotropy of FBS even when H = H a theoretically opened issue C Anisotropic field with surimposed regular texture To finish let us come back to the original issue disentangling self-similar with a true built-in anisotropy from isotropic selfsimilar processes to which an unrelated anisotropic texture is additively superimposed To address this issue let us compare a truly isotropic OSGRF X θ=α =H =5 as defined in Eq () to which a sine waveform trend with orientation θ = /8 is additively superimposed (Fig 9a) to a truly anisotropic OSGRF X θ=/8α =6H =5 The estimation and test procedures described above are applied to realizations of both processes and ˆα(θ) and H(θ) are displayed in Fig 9c and d respectively For the truly anisotropic field ( ) ˆα(θ) displays a clear minimum for ˆθ = θ with estimated anisotropy (ˆα = α(ˆθ) = 66) and selfsimilarity (Ĥ(ˆθ) = 36) parameters in close agreement with the true ones ( in 9d) This is thus clearly validating anisotropy For the isotropic field to which the directional sine wave trend has been added α(θ) shows no clear minimum and instead a rather constant behavior in θ is observed thus leading to conclude that anisotropy clearly visible on the sample field is superimposed on rather than built-in self-similarity This example leads us to conclude that the procedure proposed in the present contribution provides practitioners with a reliable tool to analyze self-similarity in presence of anisotropy and enables them to clearly disentangle built-in anisotropy from independent and superimposed added anistropic trends

10 IEEE TRANS ON IMAGE PROCESSING VI CONCLUSIONS AND PERSPECTIVES The present contribution aimed at studying images or fields where self-similarity is potentially tied to anisotropy Replacing the standard D-DWT with the HWT thus permitting to use different dilation factors along horizontal and vertical directions enabled us first to estimate the rotation and anisotropy parameters In turn this permitted a correct estimation of the self-similarity parameter along the estimated anisotropy direction This direction selection would not be permitted by the use of the sole D-DWT coefficients and thus constitutes the major benefits of the use of the HWT and therefore the key feature of the present contribution Additionally bootstrap based procedures performed in the HWT coefficient domain furnish confidence intervals for the estimates and an isotropy test that can be applied to a single image Though studied in depth for a specific Gaussian self-similar model the proposed analysis is shown to enable the detection of anisotropy for a large variety of classes of Gaussian self-similar processes Also true built-in anisotropy is clearly discriminated from isotropy to which an anisotropic trend is added Extensions of the applicability of the present method or further developments geared towards the analysis of more general classes of processes modeling Textures with scale invariance that are not necessarily exactly self-similar and that may weakly or significantly depart from Gaussian distributions are under current investigations Notably this study paves the way toward the far more difficult topic of multifractal analysis and formalim in presence of anisotropy to which future efforts are devoted MATLAB routines designed by ourselves implementing field synthesis and parameter estimation and test will be made publicly available at the time of publication REFERENCES [] S G Roux A Arneodo and N Decoster A wavelet-based method for multifractal image analysis iii applications to high-resolution satellite images of cloud structure European Physical Journal B vol 5 no pp Jun [] P Frankhauser L approche fractale : un nouvel outil dans l analyse spatiale des agglomerations urbaines Population vol pp [3] T Lundahl W J Ohley S M Kay and R Siffert Fractional brownian motion: A maximum likelihood estimator and its application to image textures IEEE Trans Medical Imaging vol 5 no [] P Kestener J-M Lina P Saint-Jean and A Arneodo Wavelet-based multifractal formalism to assist in diagnosis in digitized mammograms Image Anal Stereol vol pp 69 7 [5] M Rachidi F Richard H Bierme C Roux P Fardellone E Lespessailles C Chappard and C Benhamou Osteoporosis risk assessment: A composite index combining clinical risk factors and biophysical parameters Journal of Bone and Mineral Research vol 3 pp S S Sep 8 [6] F Richard and H Bierme Statistical tests of anisotropy for fractional brownian textures application to full-field digital mammography Journal of Mathematical Imaging and Vision vol 36 no 3 pp 7 Mar [7] M Bergounioux and L Piffet A second-order model for image denoising Set Valued and Variational Analysis vol 8 no 3- pp [8] D Schertzer and S Lovejoy Physically based rain and cloud modeling by anisotropic multiplicative turbulent cascades J Geophys Res vol 9 pp [9] P Abry J S and H Wendt When Van Gogh meets Mandelbrot: Multifractal classification of painting textures Signal Processing to appear [] M Unser Texture classification and segmentation using wavelet frames IEEE Transactions on Image Processing vol no pp [] M Nielsen L K Hansen P Johansen and J Sporring Guest editorial: Special issue on statistics of shapes and textures Journal of Mathematical Imaging and Vision vol 7 no pp Sep [] M Do and M Vetterli Wavelet-based texture retrieval using generalized gaussian density and kullback-leibler distance IEEE Transactions On Image Processing vol no pp 6 58 [3] M Chantler and L Van Gool Special issue on texture analysis and synthesis International Journal of Computer Vision vol 6 no - 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